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A Simple Introduction to Computable Analysis

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A Simple Introduction to Computable Analysis A Simple Intro duction to Computable Analysis Klaus Weihrauch FernUniversitat D Hagen corrected nd version July Contents Intro duction Computability on nite and innite words naming systems Computability on the real numb ers Eective representation of the real numb ers Op en and compact sets Continuous functions Determination of zeros Computation time and lo okahead on Computational complexity of real functionss Other approaches to eective analysis App endix References Intro duction During the last years an extensive theory of computability and computational complexity has b een develop ed see eg Rogers Rog Odifreddi Odi Weih rauchWei Hop croft and Ullmann HU Wagner and Wechsung WW Without doubt this Typ e theory mo dels the b ehaviour of real world computers for computations on discrete sets like natural numb ers nite words nite graphs etc quite adequately A large part of computers however is used for solving numerical problems Therefore convincing theoretical foundations are indisp ensible also for computable analysis Several theories for studying asp ects of eectivity in analysis havebeendevelop ed in the past see chapter Although each of these approaches has its merits none of them has b een accepted by the ma jority of mathematicians or computer scientists Compared with Typ e computability foundations of computable analysis havebeen neglected in research and almost disregarded in teaching Thispaperisanintro duction to Typ e Theory of Eectivity TTE TTE is one among the existing theories of eective analysis It extends ordinary Typ e computability theory and connects it with abstract analysis Its origin is a denition of computable real functions given by Grzegorczyk in Grz which is based on the denition of computable op erators on the set of sequences of natural numb ers Real numb ers are enco ded by fast with sp eed n converging Cauchy sequences of rational numb ers and these are enco ded by sequences of natural numb ers A real function is computable in Grzegorczyks sense i it can b e represented bya computable op erator on such enco dings of real numb ers In the following years this kind of computabilit y has b een investigated by several authors eg Grzegorczyk Grz Klaua Kla Hauck Hau Hau and Wiedmer Wie The computational complexity theory for real functions develop ed byKoandFriedmann KF Ko can b e considered as a sp ecial branch The study of representations ie functions from onto sets as ob jects of separate interest results in an essential generalization of Grzegorczyks original denition and admits to nd and justify natural computability denitions for functions on most of the sets used in ordinary analysis Basic concepts are explained in Weihrauchand Kreitz WK Wei KW Wei The theory has b een expanded in several pap ers by Hertling Kreitz Muller and Weihrauch ranging from top ological conside rations to investigation of concrete computational complexity WK KW Mue Mue WK Wei Wei Wei A Wei B HW As an interesting feature continuitycanbeinterpreted in this context as a very fundamental kind of eectivity or constructivity and simple top ological considerations explain a number of well known observations from eective analysis very satisfactorily This pap er is not a complete presentation of TTE but only a technically and con ceptually simplied selection from Wei The main stress is put on basic concepts and on simple but typical applications while the theoretical background is reduced to the bare essentials We assume that the reader has some basic knowledge in computability theory Tu ring machines computable functions recursive sets recursively enumerable sets Intro duction There are several go o d intro ductions eg the classical b o ok by Hop croft and Ull man HU or Bridges Bri In addition to standard Calculus we use some simple concepts from top ology top ological space op en and closed sets continuous func tions metric space Cauchy sequence compact set Anyintro duction to top ology eg Eng may b e used as a reference In the following we axplain some notations which will b e used in this pap er By f X Y wedenotea partial function from X to Y ie a function from a subset of X called the domain of f domf to Y The function f X Y is totalidomf X in this case we write f X Y as usual A nite alphab et is a nonempty nite set In Section denotes any nite alphab et with f g In the following section is some xed suciently large nite alphab et containing all the symb ols we shall need Let f g b e the set of natural numb ers As usual is the set of all nite words a a with k and a a The k k emptyword is denoted by Let fa a j a g fp j p g b e the i set of innite sequences or sequences with elements from We use suggestive informal notations for dening nite and innite sequences over If u a a k v b b and p c c a b c then uv a a b b l i i i k l m up a a c c u a a a a a a m times u k k k k uuu a a a a Ifx uv w and q uv p then u k k is a prex and v is a subword of x and q We extend the ab ove notations to sets of fx j is a prex of xg and nite or innite sequences For example u fp j u is a subword of pg In Chapter we generalize computability from nite to innite sequences of sym b ols and illustrate the denition bya numb er of examples Weintro duce the Cantor top ology and show that computable functions are continuous Weintro duce nota tions and representations and dene how top ological and computational concepts are transferred from sequences to named sets In Chapter weintro duce standard representations of the real numb ers the interval representation and the Cauchy re presentation and investigate the computability concepts induced by them on the real numbers Wegive examples for computable and noncomputable real numb ers wecharacterize the recursively enumerable subsets of IR and prove computabilityof anumb er of real functions In Chapter we give reasons for selecting the interval and the Cauchy representation and for rejecting eg the decimal representation W eprove that every computable real function is continuous weformulate the the sis that every physically computable function is continuous and we provethatno injective and no surjective representation can b e equivalent to the Cauchy represen tation In Chapter weintro duce representations of the op en and of the compact subsets of the real numb ers We prove eectiveversions of some well known classical prop erties esp ecially weprove a computational version of the HeineBorel theorem on compact sets Weintro duce representations of the classes C IRand C of continuous real functions and discuss their eectivity in Chapter We presentsome computational versions of well known prop erties and consider the determination of a mo dulus of continuity of the maximum value the derivative and the integral Determination of zeros of continuous functions is considered in Chapter Weprove that the general problem can not even b e solved continuously Under certain restric tions wehave a computable but nonextensional solution op erator A computable op erator exists only on the set of continuous functions whichhave exactly one zero In Chapter weintro duce as new concepts computation time and input lo okahead of Typ e machines with innite output In Chapter wedenethemodiedbinary representation which is appropriate for intro ducing computational complexity of real functions We determine b ounds of time and input lo okahead for addition multipli cation and by an application of Newtons metho d for inversion Finally we dene the complexity of compact sets which can b e interpreted as plotter complexity Some other approaches to eective analysis are discussed in Chapter Computability on Finite and Innite Words Naming Systems Computability on Finite and Innite Words Naming Systems In this Chapter is any nite alphab et ie any nite non emptysetTuring machi nes are a convenient mathematical mo del for dening computabilityofwordfunctions k f By the ChurchTuring thesis a word function is computable informally or bya physical device if and only if it can b e computed bya Turing machine Moreover Turing machines mo del time and storage complexityofphysical computers rather realistically A standard reference is the b o ok by Hop croft and Ullman HU In this section weintro duce our basic computational mo del for computable analysis the Type machinesWeformulate a generalization of the ChurchTuring thesis we prove that computable functions on nite or innite sequences are continuous and dene recursively enumerable sets Weintro duce notations and representations and dene how eectivity of elements sets functions and relations is transferred by naming systems Many examples illustrate the denitions Roughly sp eaking a Typ e machine is a Turing machine for which not only nite but also innite sequences of symb ols may b e considered as inputs or outputs We give an informal denition of Typ e machines and their semantics Denition Typ e machines ATyp e machine M is dened bytwo comp onents i a Turing machine with k oneway input tap es k a single oneway output tap e and nitely manywork tap es ii a typ e sp ecication Y Y Y with fY Y gf g k k The typ e sp ecication expresses that f Y Y Y is the typ e of M k the function computed by the machine M It tells which of the input and output tap es are provided for nite and which for innite sequences Notice that input and output tap es are restricted to oneway left to right
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